It can be useful to know that here natural units are used ([itex]c=1[/itex]) and that the masses of the neutrino are considered much smaller than their momenta ([itex]m << p[/itex])
Still, I can't understand where the [itex]\frac{m^2_i - m^2_j}{2p}[/itex] comes from.

I don't think it's a symmetry...
I think it has to do with the fact that the momentum is described by the flavor and not by the mass eigenstates...
in other words, when you expand a flavor eigenstate:
[itex] v_{f}[/itex] it has to have some momentum [itex]p[/itex]
then the expanded ones should keep the same momentum...and all the differences are supposed to come from the masses